Lower estimate of the isoperimetric deficit of convex domains inR n in terms of asymmetry

For convex bodiesD inRⁿ it is shown that the isoperimetric deficit ofD is minorized by a constant times the square of thebarycentric asymmetry (D) ofD. Here (D) is defined as the volume ofDB_D divided by the volume ofD, whereB_D denotes the ball centred at the barycentre ofD and having the same volume asD.Dedicated to the memory of Børge Jessen